This document proposes a method for optimally placing a jammer node in a wireless localization network to degrade the accuracy of locating target nodes. It formulates the problem of maximizing the minimum Cramér-Rao lower bound (CRLB) for target nodes as an optimization with constraints on jammer location. Theoretical results show the optimal jammer placement is determined by one, two or three target nodes depending on scenarios. Explicit expressions are derived for optimal placement with two targets. Numerical examples illustrate key results.

1. Optimal Jammer Placement in Wireless Localization Networks
Sinan Gezici∗
, Suat Bayram♭
, Mohammad Reza Gholami♮
, and Magnus Jansson♮
∗ Department of Electrical and Electronics Engineering, Bilkent University, 06800, Ankara, Turkey
♭ Department of Electrical and Electronics Engineering, Turgut Ozal University, 06010, Ankara, Turkey
♮ ACCESS Linnaeus Centre, Department of Signal Processing, KTH–Royal Institute of Technology, Stockholm, Sweden
Abstract
The optimal jammer placement problem is proposed for a wireless localization network, where
the aim is to degrade the accuracy of locating target nodes as much as possible. In particular, the
optimal location of a jammer node is obtained in order to maximize the minimum of the Cram´er-
Rao lower bounds for a number of target nodes under location related constraints for the jammer
node. Theoretical results are derived to specify scenarios in which the jammer node should be
located as close to a certain target node as possible, or the optimal location of the jammer node is
determined by two or three of the target nodes. In addition, explicit expressions for the optimal
location of the jammer node are derived in the presence of two target nodes. Numerical examples
are presented to illustrate the theoretical results.
Introduction
• Position information: vital for many location aware services/applications
• The main aim in wireless localization networks: achieving high localization accuracy
• Wireless sensor localization: a well investigated topic
• The effects of jamming on wireless localization networks: a rather new topic - little
attention in the literature
• Recent work [1]: the optimal power allocation strategies for the jammer nodes in or-
der to maximize the average or the minimum Cramr-Rao lower bounds (CRLBs) of
the target nodes.
• Current work: optimal jammer placement problem based on maximization the mini-
mum CRLB
System Model
• NA anchor nodes at known locations yi ∈ R2
and NT target nodes at xi ∈ R2
• A jammer node at z ∈ R2
• The jammer node is assumed to transmit zero-mean Gaussian noise
• Ai {j ∈ {1,...,NA} | anchor node j is connected to target node i}, i ∈ {1,...,NT}
Received signal: ri j(t) =
Lij
∑
k=1
αk
i js(t −τk
i j)+γi
√
PJ vi(t)+ni j(t), t ∈ [0,Tobs], j ∈ Ai
ni j(t),
√
PJ vi(t): independent zero-mean white Gaussian random processes
• τk
i j
yj−xi +bk
ij
c bk
i j ≥ 0 & Ai A L
i ∪A NL
i
• bi j [b2
i j ...b
Lij
i j ] if j ∈ A L
i & bi j [b1
i j ...b
Lij
i j ]T
if j ∈ A NL
i
• θi [xT
i bT
iAi(1) ··· bT
iAi(|Ai|)]T
• CRLB: E{ ˆxi −xi
2
} ≥ tr{[F−1
i ]2×2} with [F−1
i ]2×2 = Ji(xi,PJ)−1
• Ref.[2] : Ji(xi,PJ) = ∑
j∈A L
i
λij
N0/2+PJ|γi|2
φijφT
ij λi j
4π2
β2
|α1
ij|2 ∞
−∞ |S( f)|2
d f
c2 (1−ξj)
φi j [cosϕi j sinϕi j]T
β =
∞
−∞ f2|S( f)|2d f
∞
−∞ |S( f)|2d f 0 ≤ ξj ≤ 1: the path-overlap coefficient
ϕi j: the angle between target node i and anchor node j
• CRLBi = tr{Ji(xi,PJ)−1
} = ri(PJ|γi|2
+N0/2) ri tr{[∑j∈A L
i
λi jφi jφT
i j]−1
}
Optimal Jammer Placement
Generic Formulation and Analysis
maximize
z
min
i∈{1,...,NT }
ri PJ|γi|2
+
N0
2
subject to z−xi ≥ ε , i = 1,...,NT
Assumption: |γi|2
= ˜Ki
d0
z−xi
ν
for z−xi > d0
maximize
z
min
i∈{1,...,NT }
ri
KiPJ
z−xi
ν
+
N0
2
subject to z−xi ≥ ε , i = 1,...,NT
(1)
Ki ˜Ki(d0)ν
Proposition 1: If there exists a target node that satisfies the following inequality,
rℓ
KℓPJ
εν
+
N0
2
≤ min
i∈{1,...,NT }
i=ℓ
ri
KiPJ
( xi −xℓ +ε)ν
+
N0
2
(2)
and if set {z : z−xℓ = ε & z−xi ≥ ε, i = 1,...,ℓ−1,ℓ+1,...,NT} is
non-empty, then the solution of (1), denoted by zopt
, satisfies zopt
−xℓ = ε; that is,
the jammer node is placed at a distance of ε from the ℓth target node.
The optimization problem in (1) in the presence of two target nodes ℓ1 and ℓ2 only:
maximize
z
min
i∈{ℓ1,ℓ2}
ri
KiPJ
z−xi
ν
+
N0
2
subject to z−xℓ1
≥ ε , z−xℓ2
≥ ε
(3)
ℓ1,ℓ2 ∈ {1,...,NT} and ℓ1 = ℓ2. zopt
ℓ1,ℓ2
and CRLBℓ1,ℓ2
: the optimizer and the optimal
value of (3)
Proposition 2: Let CRLBk,i be the minimum of CRLBℓ1,ℓ2
for ℓ1,ℓ2 ∈ {1,...,NT} and
ℓ1 = ℓ2, and let zopt
k,i denote the corresponding jammer location (i.e., the optimizer of
(3) for ℓ1 = k and ℓ2 = i). Then, an optimal jammer location obtained from (1) is
equal to zopt
k,i if zopt
k,i is an element of set z : z−xm ≥ ε, m ∈ {1,...,NT}{k,i} and
rm
KmPJ
zopt
k,i −xm
ν
+
N0
2
≥ CRLBk,i (4)
for m ∈ {1,...,NT}{k,i}.
⇒ the optimal jammer location is mainly determined by two of the target nodes since
the others have larger CRLBs when the jammer node is placed at the optimal location
according to those two jammer nodes only.
——————————–Special cases———————————–
1-Two Target Nodes
Proposition 3: For the case of two target nodes (i.e., NT = 2), the solution zopt
of (1)
satisfies one of the following conditions:
(i) if x1 −x2 < 2ε, then zopt
−x1 = zopt
−x2 = ε
(ii) otherwise,
(a) if r1
K1PJ
εν + N0
2 ≤ r2
K2PJ
( x1−x2 −ε)ν + N0
2 , then zopt
−x1 = ε and
zopt
−x2 = x1 −x2 −ε
(b) if r2
K2PJ
εν + N0
2 ≤ r1
K1PJ
( x1−x2 −ε)ν + N0
2 , then zopt
−x1 = x1 −x2 −ε and
zopt
−x2 = ε
(c) otherwise, zopt
−x1 = d∗
and zopt
−x2 = x1 −x2 −d∗
, where d∗
is the unique
solution of the following equation over d ∈ (ε, x1 −x2 −ε).
r1
K1PJ
dν
+
N0
2
= r2
K2PJ
( x1 −x2 −d)ν
+
N0
2
(5)
2-Infinitesimally Small ε
Proposition 4: Suppose NT ≥ 3 and ε → 0. The max-min CRLB in the presence of
target nodes ℓ1, ℓ2, and ℓ3: CRLBℓ1,ℓ2,ℓ3
= maxz minm∈{ℓ1,ℓ2,ℓ3} rm
KmPJ
z−xm
ν + N0
2 . Also,
let target nodes i, j, and k achieve the minimum of CRLBℓ1,ℓ2,ℓ3
for
ℓ1,ℓ2,ℓ3 ∈ {1,...,NT } and let zopt
i, j,k denote the jammer location corresponding to
CRLBi,j,k. Then, the optimal location for the jammer node (i.e., the optimizer of (1) in
the absence of the distance constraints) is equal to zopt
i, j,k, and at least two of the CRLBs
of the target nodes are equalized by the optimal solution.
Numerical Examples
• Simulation parameters: ε = 1m, PJ = 6, N0 = 2, ν = 2, and Ki = 1 for i = 1,...,NT.
LOS scenarios, λi j = 100N0 xi −yj
−2
/2 (free space propagation model)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Optimal location of the jammer node
x [m]
y[m]
Anchor node
Target node
Target 1
Target 2
Target 3
Target 4
ε
ε
ε
ε
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Optimal location of the jammer node
x [m]
y[m]
Anchor node
Target node
Target 1
Target 2
Target 3
Target 4
ε
ε
ε
ε
Fig. 1. The network consisting of anchor nodes at [0 0], [10 0], [0 10], and [10 10] m. Left figure: target nodes at [2 6], [5 2], [8 3], and [9 6] m.
Right figure target nodes at [2 5], [4 1], [8 8], and [9 2] m.
V. N E• Left Fig: Proposition 2 for k = 1&i = 3 → the solution determined by subnetwork
consisting of target 1 and 3. Solution: zopt
1,3 = [5.0605 4.4697] m, CRLB1,3 = 0.8053m2
• Right Fig: Subnetwork consisting of taget nodes 1,3, and 4 archives the minimum
max min CRLB among all other possible subnetworks with three target nodes.
Solution: CRLB1,3,4 = 0.7983m2
and zopt
1,3,4 = [5.5115 5.5717]m
References
[1] S. Gezici, M. R. Gholami, S. Bayram, and M. Jansson, “Optimal jamming of wireless localization
systems,” in IEEE International Conference on Communications (ICC) Workshops, June 2015.
[2] Y. Shen, and MZ. Win, “Fundamental limits of wideband localizationPart I: A general framework,”
IEEE Transactions on Information Theory, pp. 4956-4980, 2010.
Emails: gezici@ee.bilkent.edu.tr, sbayram@turgutozal.edu.tr, mohrg@kth.se, janssonm@kth.se